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Tuesday, October 23, 2012

Definition — a precise and unambiguous description
of the meaning of a mathematical term. It characterizes the meaning of a
word by giving all the properties and only those properties that must
be true.Theorem — a mathematical statement that is proved
using rigorous mathematical reasoning. In a mathematical paper, the
term theorem is often reserved for the most important results.Lemma — a minor result whose sole purpose is to help
in proving a theorem. It is a stepping stone on the path to proving a
theorem. Very occasionally lemmas can take on a life of their own (Zorn’s lemma, Urysohn’s lemma, Burnside’s lemma, Sperner’s lemma).Corollary — a result in which the (usually short)
proof relies heavily on a given theorem (we often say that “this is a
corollary of Theorem A”).Proposition — a proved and often interesting result, but generally less important than a theorem.Conjecture — a statement that is unproved, but is believed to be true (Collatz conjecture, Goldbach conjecture, twin prime conjecture).Claim — an assertion that is then proved. It is often used like an informal lemma.Axiom/Postulate — a statement that
is assumed to be true without proof. These are the basic building blocks
from which all theorems are proved (Euclid’s five postulates, Zermelo-Fraenkel axioms, Peano axioms).Identity — a mathematical expression giving the equality of two (often variable) quantities (trigonometric identities, Euler’s identity).Paradox — a statement that can be shown, using a
given set of axioms and definitions, to be both true and false.
Paradoxes are often used to show the inconsistencies in a flawed theory
(Russell’s paradox). The term paradox is often used informally to
describe a surprising or counterintuitive result that follows from a
given set of rules (Banach-Tarski paradox, Alabama paradox, Gabriel’s horn).